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Open Access Research A stochastic model of oncogene expression and the relevance of this model to cancer therapy Francis D Alfano* Address: The Harold Leever Cancer Center, 1075 Chase Pa

Open Access

Research

A stochastic model of oncogene expression and the relevance of this model to cancer therapy

Francis D Alfano*

Address: The Harold Leever Cancer Center, 1075 Chase Parkway, Waterbury, Connecticut, 06708, USA

Email: Francis D Alfano* - fralfano@aol.com

* Corresponding author

Abstract

Background: Ablation of an oncogene or of the activity of the protein it encodes can result in

apoptosis and/or inhibit tumor cell proliferation Therefore, if the oncogene or set of oncogenes

contributing maximally to a tumor cell's survival can be identified, such oncogene(s) are the most

appropriate target(s) for maximizing tumor cell kill

Methods and results: A mathematical model is presented that describes cellular phenotypic

entropy as a function of cellular proliferation and/or survival, and states of transformation and

differentiation Oncogenes become part of the cellular machinery, block apoptosis and

differentiation or promote proliferation and give rise to new states of cellular transformation Our

model gives a quantitative assessment of the amount of cellular death or growth inhibition that

result from the ablation of an oncogene's protein product We review data from studies of chronic

myelogenous leukemia and K562 cells to illustrate these principles

Conclusion: The model discussed in this paper has implications for oncogene-directed therapies

and their use in combination with other therapeutic modalities

Background

For the past thirty years, cancer research has elucidated a

family of genes that are integrally involved in the cancer

process These genes comprise two subsets One subset,

termed oncogenes, gives rise to proteins that modulate

such processes as cell cycle progression, signaling, cellular

growth and apoptosis [1-3] The other consists of genes

that can suppress tumor activity and their absence can

lead to the initiation and/or progression of cancer [4]

There has been a body of work discussing the view that

cancer arises from genetic instability Evidence for genetic

instability that has been cited includes the occurrence of

chromosomal abnormalities, microsatellite DNAs and

aberrant gene expression through hypermethylation of

DNA [5-7] Moreover, recent work using RNA microarray

analysis has shown that there are key genes that are over-expressed as a result of malignant transformation, and others that are under expressed, compared to RNA tran-scripts in nonmalignant counterparts [8] Many of these gene expression changes illustrated by gene microarray analysis may be secondary or even far distal to the primary changes determined by the actual oncogene or suppressor gene In order to define the complex behavior of a tumor cell population associated with these complex gene expressions, we have chosen to define entropies of apop-tosis, cellular differentiation and survival or growth inhi-bition We hypothesize that a cell's phenotypic entropy is determined as a function of the survival fraction or prolif-eration rate of a tumor ;and also, the number of trans-formed and differentiated states that arise within a

Published: 31 January 2006

Theoretical Biology and Medical Modelling 2006, 3:5 doi:10.1186/1742-4682-3-5

Received: 02 December 2005 Accepted: 31 January 2006 This article is available from: http://www.tbiomed.com/content/3/1/5

© 2006 Alfano; licensee BioMed Central Ltd

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

particular cell population The mathematical relations

that we have formulated can quantitatively determine

how ablation of an oncogene's protein activity can result

in apoptosis and/or a decrease in proliferation within a

population of tumor cells The goal is then to determine

which oncogene or set of oncogenes contributes

maxi-mally to a tumor cell's survival; and thereby, to predict

which oncogene(s) are the most appropriate target(s) for

maximizing tumor cell kill

The model

The cellular phenotypic entropy is determined by first

defining all allowable phenotypes These phenotypes

include the set of transformed states associated with

aber-rant gene expression, the set of differentiated states that

have been defined as a result of stochastic gene expression

and the distribution of cells between living and dead or

growth inhibited Therefore

fs Ent(phenotype) = Ent(transform/differentiation) fs +

Ent(cell survival) (1)

where Ent(transform/differentiation) is the entropy of

transformation and differentiation per observed

popula-tion and Ent(cell survival) is the entropy of cell death or

growth inhibition per total population; fs is the ratio of

observed cells to predicted cells

The transformed states are the phenotypes that a cell can

access which provide a hyperproliferative advantage over

the cell's normal counterpart This includes phenotypes

that have a better growth advantage as well as phenotypes that have antiapoptotic behaviors in environments that would normally lead to cellular apoptosis The total number of states that a cell can access is determined by the number of end-differentiated states and by the number of new "environments" that a transformed cell can inhabit over and above its normal counterpart Let Ω represent the number of transformed states that a cell can access as a transformed cell and let ω represent the number of states that the cell's normal counterpart can access via differen-tiation We presume that ω is equal to the number of end-differentiated states in an unend-differentiated cell or equal to

1 in an end-differentiated cell We will define the entropy

of the combined states of transformation and differentia-tion as

fs Ent(transform/differentiation) = c fs ln(ω Ω + ω) (2) Equation (2) is motivated by the classic definition of entropy as applied to biological systems [9] If a system can exist in N equivalent configurations, then the entropy

of that system is given by Entropy = c ln N

where "c" is a constant of proportionality

In equation (2), we presume that each purely transformed state will ultimately interact with each differentiated state giving rise to a new and unique transformed state For example, an undifferentiated cell containing a trans-formed phenotype and differentiating into two differenti-ated cells will occupy one of two possible transformed phenotypes

Each transformed state confers a survival advantage to the transformed cell as compared with the cell's normal coun-terpart More practically, transformed cells are also defined by the set of oncogene/suppressor genes that are active within the cell and the number of different cellular mechanisms that this set of genes acts upon to change the cell's behavior with respect to growth and apoptosis To simplify further analysis, let us consider the action of oncogenes only and disregard the action of suppressor genes Also, we will assume that there is a direct correla-tion between the set of 'environments' that a transformed cell can inhabit and the unique cellular mechanism that

an oncogene affects Therefore, let us correlate each trans-formed state Ω with the number of oncogenes multiplied

by the unique cellular mechanisms that each oncogene affects (see Fig 1)

The term Ent(cell survival) in equation (1) can be calcu-lated by defining cell death (or decreased proliferation) and cell viability (or enhanced viability) as two states that

Oncogenes can affect multiple cellular pathways, which result

in modulating proliferation and apoptosis

Figure 1

Oncogenes can affect multiple cellular pathways, which result

in modulating proliferation and apoptosis Depicted here are

three oncogenes Oncogene 2's activity overlaps with the

activities of the other two Therefore, ablation of oncogene

2's activity would not result in any measurable change in

pro-liferation or apoptosis

are independent of the number of transformed states but

still nonetheless contribute to the overall phenotypic

entropy Define the total number of cells, which is

deter-mined by a suitable control, as Nc and the measured

number of observed cells as ns The difference, Nc-ns, in

some cases would represent apoptosis or cell death, and in

other cases would represent decreased proliferation

meas-ured against a suitable standard If Ent(cell survival) is

defined in a canonicalthermodynamic formalism [9],

then

Nc Ent(cell survival) = c ln(Nc!/ns!(Nc-ns)!) (3)

Utilizing Stirling's approximation [10] and defining fs =

ns/Nc, Equation (3) becomes

Ent(cell survival) - c [(1-fs) ln(1-fs) +fs ln fs] for 0<fs<1

(4)

and

Ent(cell survival) = 0 for fs = 0 or 1 by continuity

Combining equations (1), (2) and (4) gives us

fs [Ent(phenotype)] = - c [(1-fs) ln(1-fs) +fs ln fs] + c fs ln(ω

Ω + ω) or

Ent(phenotype) = - c [((1-fs)/fs)ln(1-fs)+ ln fs] + c ln(Ω +

1)+ c ln(ω) (5)

Equation (5) is the general statement of the model, which

relates the phenotypic entropy to the processes of

differ-entiation, transformation and cellular growth and/or

apoptosis

Since the entropy in equation (5) is defined along the

clas-sic definition of entropy, we can apply the second law of

thermodynamics to our analysis Consider a modulator of

differentiation and/or oncogene activity that either

reduces or completely eliminates the action of the

onco-gene or changes the number of available differentiated

phenotypes We know that Ent(phenotype) should

increase or remain equal with time, and therefore this

entropy, after administration of the oncogene modulator,

should be greater than or equal to the entropy prior to its

administration However, if the inhibitor is removed,

then the entropy after removal should return to that of the

premodulator's environment, i.e

Ent(phenotype)premodulator ≤ Ent(phenotype)postmodulator ≥

Ent(phenotype) premodulator (6)

The result is self-consistent only if equation (6) is

consid-ered with the equal signs Therefore, if we take times pre

and post modulator administration that most closely approximate the equality of entropies pre and post the administration of the modulator, then

{- [((1-fs)/fs) ln(1-fs)+ ln fs] + ln(Ω + 1)+ ln(ω)}premodulator

= {- [((1-fs)/fs) ln(1-fs) + ln fs] + ln(Ω + 1)+ ln(ω)}post_modulator (7)

In Table 1, we consider multiple examples where changes

of Ω and ω occur as a result of the modulator; fs (pre mod-ulator) is for most of the examples taken as 1 but we con-sider examples of modulator given in the setting of cytotoxic drugs, which can reduce fs (premodulator) to less than 1 The intent of these substitutions is to calculate

fs (post modulator) to determine the effect of the modula-tor in different settings We can solve for fs (post

modula-tor) in equation (7) by using the root function of

MATHCAD version 11 [25]

Chronic mylogenous leukemia and K562

Our intent is to study oncogenic behavior in realistic models to determine whether the principles outlined above can predict outcomes of therapy As an example, let

us consider the Bcr-Abl oncogenic protein, which is the transforming agent for chronic myelogenous leukemia (CML) [11,12] The Bcr-Abl protein is the result of the fusion of sequences from the Abl proto-oncogene on chromosome 9 with the sequences from the proto-onco-gene, Bcr, on chromosome 22 The two major forms of Bcr-Abl, p210 and p190, can each cause chronic myeloge-nous leukemia (CML) in humans The Abl component of this protein encodes a nonreceptor tyrosine kinase that is constitutively active and activates a number of signal transduction pathways involved with cell proliferation and apoptosis Bcr-Abl can inhibit apoptosis and decrease cell proliferation by its kinase action in experimental sys-tems and myeloid cells These mechanisms have been shown to be mediated for the most part through (1) acti-vation of phosphatidylinositol 3-kinase (PI-3K) and (2) Jak-Stat kinases In addition, Bcr-Abl can affect p53 and MYC in a RAS-dependent manner [12-14] and can acti-vate Jun N-terminal kinase (JNK)

In CML, Marley et al [15,16] found that the antiprolifera-tive effect of the Bcr-Abl inhibitor Imatinib correlated most closely with the inhibition of PI-3K within chronic myeloid leukemia progenitor cells, and also found that AG490, a Jak2 kinase inhibitor and FTI II, a farnesyltrans-ferase inhibitor and an inhibitor of RAS activation, could also reduce the proliferation of clonogenic CML cells This suggests that Bcr-Abl can influence at least three separate proliferation or antiapoptotic mechanisms within CML cells and effect transformation by activating three separate cellular mechanisms (see Fig 2) There is evidence that CML is a disease of stem cells that can undergo

self-renewal as well as differentiate into committed progenitor

cells capable of proliferating Laboratory evidence has

shown that drugs such as interferon or Imatinib, the

inhibitor of the Bcr-Abl kinase, have different

antiprolifer-ative effects on CML stem cells and committed progenitor

cells [17-19] Therefore, any analysis of CML proliferation

and apoptosis needs to take into account these two

dis-tinct cellular types

Imatinib is a tyrosine kinase inhibitor that specifically

binds the ATP pocket of Bcr-Abl tyrosine kinase,

inhibit-ing the activity of the kinase Moreover, it is known to

induce apoptosis in Bcr-Abl positive cells [15,16] In CML

cells, the predominant effect of Imatinib is not to induce

apoptosis but to decrease proliferation of the committed

progenitor cells and to a lesser extent the CML stem cells

[17,19] We can use equation (7) and Table 1 to calculate

the effect of Imatinib on the stem cell and committed

pro-genitor populations The committed propro-genitor

popula-tion is a differentiated system, and therefore

ω(preImatinib) = ω(postImatinib) The term [(1-fs)/fs

ln(1-fs) +ln fs]preImatinib is taken to be zero since fs

(preImat-inib) is taken to be nearly equal to one Since the Bcr-Abl

kinase predominantly affects three enzyme mechanisms,

the Ω(preImatinib) is equal to three We will assume the

maximum effect of Imatinib and so take Ω(postImatinib)

to be zero Therefore, symbolically, we have

{Ω(Pre)->Ω(Post): 3->0; ω(Pre)->ω(Post): 1->1; fs(Pre) = 1} By

referring to Table 1, we find fs(Post) to be 0.5 In CML

stem cells, Imatinib has been shown to have much less of

an impact on cellular proliferation One postulated

mech-anism for this is the presence of an enhanced multidrug

resistance protein (MDR) which extrudes the drug from

the interior of the cell [20] Therefore, Imatinib may not

maximally inhibit the cellular mechanisms outlined

above We can reasonably postulate that

Ω(Pre)->Ω(Post): 3->1 Furthermore, the effect of Imatinib on

dif-ferentiation of the CML stem cell is not clear Schuster et

al [21] were able to show that the block of differentiation

on a murine hematopoietic progenitor line by Bcr-Abl kinase was reversed by Imatinib but Angstreich et al [22] were unable to show any effect on the differentiation of CML progenitor stem cells by Imatinib For our analysis of CML stem cells, we will consider a mixing of two states; i.e {Ω(Pre)->Ω(Post): 3->1; ω(Pre)->ω(Post): 2->2;

fs(Pre) = 1} and {Ω(Pre)->Ω(Post): 3->1;

ω(Pre)->ω(Post): 2->1; fs(Pre) = 1} to reflect this duplicity of dif-ferentiation data Table 1 shows that fs = 0.77 and 0.5 for these two situations, respectively This establishes the range of fs to be between 0.5 and 0.77 for CML stem cells Holtz et al [19] studied the separate effects of Imatinib on CML stem cells and committed progenitor cells and derived an index of inhibition that is appropriate for our analysis They found a progenitor frequency that was decreased by 52 ± 5 % for committed progenitors and 43

± 12% for primitive progenitors (stem cells) If we associ-ate one minus the percent decrease in progenitor fre-quency with fs, then fs is equal to 0.48 ± 0.05 and 0.57 ± 0.12 by their data, in agreement with our theoretical pre-dictions

Imatinib and the chemotherapy drug cytosine-arabino-side have been shown to induce apoptosis and erythroid differentiation in the Bcr-Abl positive cell line K562 [23] Fang et al [23] also demonstrated a strong influence by Imatinib on the Akt kinase system of K562 cells Imatinib induced erythroid differentiation in 37.5 % of K562 cells Arnaud et al [24] also observed erythroid differentiation

in response to Imatinib and furthermore observed meg-akaryocytic differentiation with respect to phorbol esters Therefore, for the K562 system, one can consider a mix of the states {Ω(Pre)->Ω(Post): 3->2; ω(Pre)->ω(Post): 3->2;

fs(Pre) = 1} and {Ω(Pre)->Ω(Post): 3->2;

ω(Pre)->ω(Post): 3->3; fs(Pre) = 1} to represent this system We can compute values of fs(Post) as 0.77 and 0.92,

respec-Table 1: Tumor cell survival fraction as a function of changes in the states of transformation, differentiation, and the initial tumor survival fraction.

The term fs(Pre) is the survival fraction of a tumor cell's population prior to the application of a targeted therapy; Ω(Pre)-> Ω(Post) represents the change in transformed states resulting from oncogene inhibition ; ω(Pre)->ω(Post) represents the change in differentiation states resulting from oncogene inhibition The term fs(Post) is the survival fraction or reduced proliferation of a tumor cell population that results from oncogene inhibition and is calculated from equation (7).

tively Fang et al found that the percentage of

nonapop-totic cells measured by an Annexin V assay was 81.7 ±

2.4% and by a morphology assay was 84.9 ± 1.6%

When cytosine-arabinoside was added to the system,

dif-ferentiation remained about the same at 38.8% but the

percent of nonapoptotic cells decreased to 71.2 ± 1.8%

and 65 ± 0.3% by the Annexin and morphology assays,

respectively Since cytosine-arabinoside alone induced an

apoptosis of 15%, we have that fs(PreImatinib but in the

presence of cytosine-arabiniside) was0.85 Substituting

this into the above states, we have {Ω(Pre)->Ω(Post):

3->2; ω(Pre)->ω(Post):3->3; fs(Pre) = 85} and

{Ω(Pre)->Ω(Post): 3->2; ω(Pre)->ω(Post): 3->2; fs(Pre) = 85} By

referring to Table 1, we find that fs(Post) is equal to 0.74

and 0.57, respectively The theoretical values are within

the range of the experimental data

Discussion

We have developed a model of cellular behavior that

interprets cellular transformation, apoptosis/proliferation

and differentiation as stochastic processes The model

defines the necessity of considering all these mechanisms

of cellular behavior together because there is

interdepend-ence amongst them For example, differentiation may

lead to the generation of apoptosis or decreased cellular

proliferation; and transformation can result in enhanced

proliferation when compared to the transformed cell's

normal counterpart Others have interpreted cellular

transformation as a stochastic process [9,26] and several

lines of evidence have been developed to explain the

underlying cause of the stochastic behavior of cancer Genetic instability as a cause for cancer has been a recur-ring theme since the classic paper of Boveri [27] and is defined by most authors as the generation of altered cellu-lar behavior because of an altered protein network sec-ondary to the introduction of a new oncogene protein or the removal of a tumor suppressor protein In either case, definite outcomes are thought to be predicted by either event Furthermore, over a long enough period of time, cellular behavior can evolve within a transformed popula-tion of cells, leading to a heterogeneous set of cellular behaviors

We have used entropy as a measure of change for transfor-mation; not only because entropy is a linear function and often different items of interest can simply be added together, but also because this approach is supported by past analyses that have used entropy to model cancer behavior in the context of chemical carcinogenesis [9] Furthermore, recent work on the differentiation of mye-loid colony-forming cells has shown that experimental data best fit a stochastic model [28] Because of the sim-plicity of the entropy function, we can collect components

of cellular behavior that best fit our knowledge of the cel-lular phenotype; i.e the cell's growth capacity and sur-vival, the cell's differentiation status and the cell's transformation status Each of these quantities can be defined within the context of an entropy function and combined to serve as an index of cellular phenotypic entropy

We consider the cellular phenotypic entropy to remain constant during therapies that are observed to be reversi-ble As an example for study, we chose chronic myeloge-nous leukemia because this model is well defined in terms

of the oncogenes involved Imatinib induces a high rate of remissions when given in a clinical context, but when Imatinib is discontinued the disease returns to a clinical state identical to that observed before the inhibitor This observation also has been made in vitro [29] Such would not be the case with most chemotherapies since they are often mutagenic and exert their effect by modifying cellu-lar DNA permanently [30]

In our analysis of Bcr-Abl kinase action in CML, we sur-mised that the protein is acting predominantly over three kinase systems to enhance cellular proliferation These three systems involve pathways that have already been elucidated such as (1) the Akt kinase system, (2) the RAS dependent p53 system, and (3) the Jak-Stat kinase system [12,13] However, these systems are not totally independ-ent and their interdependence may serve to reduce the number of transforming states that we have designated within our mathematical computations As such, the Jun pathway was also recognized to be affected by the

BCR-Bcr-Abl modulates up to four cellular pathways, but because

incurred by the oncogene's behavior

Figure 2

Bcr-Abl modulates up to four cellular pathways, but because

of the interdependence of these pathways, we conclude that

3 pathways best represent the number of transformed states

incurred by the oncogene's behavior

ABL kinase, but it is not clear that this is strongly

impli-cated in Bcr-Abl kinase action within the context of CML

Even if it were appropriate to consider the Jun pathway,

the interdependence of these pathways may still justify

equating Ω(Pre), the number of transformed states

deter-mined by the Bcr-Abl kinase, to three (see Fig 2)

By our model, each enzyme system is correlated with a

transformed state, and the more each system is affected by

the inhibitor, the greater the effect the inhibitor has on

reducing cellular growth and/or inducing apoptosis

Fur-thermore, if the inhibitor contributes to the

differentia-tion of the transformed cell, then that also will contribute

to a greater reduction of cellular proliferation and a

possi-ble increase in apoptosis In all the instances we cited, the

data supports our theoretical model

Our model does not address the issue of how a normal

cell with low phenotypic entropy becomes transformed to

a cell with higher entropy Even if Imatinib fully ablates

the activity of the Bcr-Abl kinase, the cell remains

trans-formed and the phenotypic entropy does not change

Therefore, transformation by our model is considered

independent of oncogene expression (see Figure 3) How

can this be? The answer lies in the fact that the signature

of transformation is not in the expression of the oncogene

protein but in the alteration in the DNA by the oncogene

In the case of Bcr-Abl, it is the observed 9–22

chromo-some translocation that affects the behavior of other

nor-mal cellular proteins [31] Such a conclusion is supported

by the observations of Keating et al., who observed varia-ble expression of Bcr-Abl transcripts in early CML progen-itor cells that exhibited the chromosome translocation [32]

More studies will be needed to determine whether this is

a specific feature of Bcr-Abl positive disease or a manifes-tation of a more general principle of cellular transforma-tion and cancer Namely, does an oncogene act in a similar fashion within a set of oncogenes as it does when

it acts alone? If so, then it would be important to know how to measure an oncogene's action so that one could target the specific oncogene with the greatest impact on a tumor cell's survival Within the context of our model, we can provide a recipe for calculating the extent to which inhibiting an oncogene's action can reduce tumor cell sur-vival by answering the following questions:

(1) How many proliferation/apoptosis mechanisms are active in the tumor cell's normal counterpart? (In CML,

we argued for three mechanisms.) (2) What oncogenes inhibit which mechanisms? The answer would most likely be specific to the oncogenes that are active within the tumor cell

(3) How many end-differentiated states apply to the tumor's specific environment, and does the oncogene inhibitor change the number of differentiated states expressed after oncogene ablation, and does the inhibitor completely ablate the oncogene's action with respect to all proliferation/apoptosis mechanisms? And finally, (4) What is the initial survival fraction of the tumor cell's population prior to oncogene inhibition? The initial sur-vival may vary depending upon other therapies applied such as radiation and/or chemotherapy Table (1) yields theoretical calculations of survival fractions as a function

of changes of oncogene expression, differentiation and the initial survival fraction before a targeted therapy is applied, demonstrating the synergy between oncogene specific therapies and other modalities such as chemo-therapy

Competing interests

The author(s) declare that they have no competing inter-ests

Acknowledgements

The author thanks two anonymous reviewers for constructive comments; and thanks Dr Paul Agutter for his help We also thank Dr Greg Angstreich for discussions concerning CML growth and differentiation.

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A normal cell, N, is transformed thereby increasing its

phe-notypic entropy

Figure 3

A normal cell, N, is transformed thereby increasing its

phe-notypic entropy When the oncogene inhibitor is applied to

the transformed cell, T, the cell maintains constant

pheno-typic entropy and therefore does not return to its normal

state As a result of the loss of the oncogene's protective

actions, the transformed cell, Ti, is less adapted to its

envi-ronment and undergoes either growth inhibition or

apopto-sis

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